We introduce a new augmented dual-mixed finite element method for the linear convection-diffusion equation with mixed boundary conditions. The approach is based on adding suitable residual type terms to a dual-mixed formulation of the problem. We prove that for appropriate values of the stabilization parameters, that depend on the diffusivity and the magnitude of the convective velocity, the new variational formulation and the corresponding Galerkin scheme are well-posed and a Céa estimate can be derived. We establish the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart–Thomas/Brezzi–Douglas–Marini and continuous piecewise polynomials. In addition, we develop an a posteriori error analysis of residual type. We derive a simple a posteriori error indicator and prove that it is reliable and locally efficient. Finally, we provide some numerical experiments that illustrate the performance of the method.

针对混合边界条件的线性对流扩散方程，我们引入了一种新的加重双重混合有限元方法。该方法基于在问题的双重混合公式中添加适当的残差类型项。我们证明，对于稳定参数的适当值（取决于扩散率和对流速度的大小），新的变分公式和相应的Galerkin方案是正确的，可以推导出Céa估计。当通量和浓度分别由Raviart–Thomas / Brezzi–Douglas–Marini和连续分段多项式近似时，我们确定收敛速度。此外，我们开发了残差类型的后验误差分析。我们推导出一个简单的后验误差指标，并证明它是可靠的和局部有效的。最后，我们提供了一些数值实验来说明该方法的性能。